Aspherical pro-$p$-groups

O. V. Mel'nikov

Sbornik. Mathematics, Tome 193 (2002) no. 11, pp. 1639-1670

Résumé

The notion of an aspherical pro-$p$-group is introduced. It is proved that if a group $G=F/N$ is aspherical, where $F$ is a free pro-$p$-group, then the relation $\mathbb F_p[[G]]$-module $\overline N=N/N^p[N,N]$ satisfies an assertion of the type of Lyndon's identity theorem. The finite subgroups and the centre of $G$ are described. The structure of an aspherical pro-$p$-group $G$ with a soluble normal subgroup $A\ne\{1\}$ is studied. In particular, if $A\cong\mathbb Z_p$, then $G$ contains a subgroup of finite index of the form $A\leftthreetimes W$ where $W$ is a free pro-$p$-group.